The canonical form of a nonlinear second order PDE
see: V.S.Vladimirov, A Collection of Problems on the Equations of Mathematical Physics, Springer, 1986
In your case characteristic equation is $$-x^2dxdy-ydx^2=0$$ with solutions $$x=c_1,\quad -1/x+\log(y)=c_2.$$ By a change of variables $$\xi=x,\quad\eta=-1/x+\log(y)$$ we get canonical form $$u_{\xi\eta}=\frac{{{e}^{\frac{1}{\xi }+\eta }}\, \left( {{\xi }^{2}}\, {u_{\xi }}-4 u\, {{\xi }^{2}}+{u_{\eta }}\right) }{{{\xi }^{4}}}-\frac{{u_{\eta }}}{{{\xi }^{2}}}$$